Exploration of graphs with excluded minors

TL;DR

This paper proves a constant competitive ratio for online graph exploration on minor-free graphs, extending previous results and establishing a connection with light spanners, leading to improved bounds for genus graphs.

Contribution

It introduces a novel connection between exploration algorithms and light spanners, enabling constant ratios on broader graph classes and improved bounds for genus graphs.

Findings

Constant competitive ratio on minor-free graphs

Construction of light spanners for genus graphs

Improved lightness bounds for genus g>0

Abstract

We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm Blocking and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus g>0 and recovers the known tight bound for the planar case (g=0).